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By Arnold Emch, Boulder, Colo. 
Read before the Academy, by title, at Iola, December 31, 1901. 


r r^HE present tendency of scientific specialization is generally justi- 
■*- tied ; it is in the interest of pure science. This is especially true 
of the devolopment of mathematical branches, where everything for- 
eign to the fundamental axioms is carefully discarded. A science in 
which this spirit is applied intentionally and systematically becomes 
a branch of philosophy, for instance, Grassmann's Linende Ans- 
dehnmgslehre, Lobatschesky's Geometry, v. Standt's Geometrie der 
Lage, Weierstrasse's Theory of Functions, etc. As such they are of 
the greatest importance for the rigorous development of mathemat- 
ical thought, and their value cannot be overestimated. It is a ques- 
tion, however, whether so-called pure methods are always what they 
pretend to be, and whether they are always to be recommended for 
pedagogical purposes. To take an example : Is it well to consider 
certain configurations in space in order to simplify the demonstration 
of propositions in plane geometry; or is it necessary, in order to 
be consistent, to apply only previously known propositions of plane 
geometry ? 

There is no doubt in my mind that the first can be done in a suc- 
cessful and consistent manner. In this paper I shall attempt to show 
the value of such methods for the teaching of the geometry of posi- 
tion. At this point I desire to say that descriptive geometry, as well 
as projective geometry, or the geometry of position, ought to be made 
regular courses in the mathematical departments of real universities. 
A knowledge of elementary descriptive geometry gives the student an 
invaluable power for the mastery of the more difficult problems of 
projective geometry and of higher geometry in general.f In what fol- 
lows I shall assume the knowledge of ordinary descriptive geometry. 


There are a great number of propositions in plane geometry which 
appear in their natural light when considered as projections of figures 
in space. As such they are independent of metrical relations and it 
is unnatural to prove them by equations. As an example, I may men- 
tion the homology of triangles. 

*A paper read before the American Association for the Advancement of Science, at Denver, 
Colo., August, 1901. 

t Regular courses in descriptive and projective geometry are now offered at nearly all uni- 
versities of continental Europe. 



Two triangles, ABC and A'B'C, are homologous, 

1. When the rays AA', BB', CC, joining corresponding points, are 
concurrent; or, also, 

2. When the points of intersection of corresponding sides A B and 
A'B', BC and B'C, CA and CA' are collinear. 

Each of these two definitions as a hypothesis necessitates the other 
as a thesis. 

Casey, in his Sequel to Euclid, which is very rich in beautiful ex- 
amples and propositions, but without an organism, reduces advanced 
geometry to an incoherent mass of metrical facts. To prove the 
propositions concerning homologous triangles, he introduces ratios of 
areas of triangles, and applies the theorem of Menelaos concerning 
transversals* The same method is followed in most treatises on 
plane geometry. The immortal elements of Euclid, which in them- 
selves are of rare beauty and rigor, lead very soon to sterility when 
applied to projective properties of figures. 


Fig. 1 

Homologous triangles appear in the simplest manner as plane sec- 
tions of triangular pyramids. Let V be the vertex of such a pyramid 
and ABC and A'B'C the intersections of its edges, with two oblique 
planes, P and P', respectively. Let S be the line of intersection of P 
and P'. A glance at the figure (fig. 1) shows that AB and A'B', 
BC and B'C, CA and CA meet in points of S. 

* Loc. cit,, book sixth. 


Now, two triangles, ABC and A'B'C, in which AA', BB', CO' 
produced are concurrent may always be considered as the projection 
of a triangular pyramid cut by two oblique planes in ABC and A'B'C 
Thus the foregoing proposition is established. In a similar manner 
the converse proposition may be proved. 

As a second example, let us take the proposition concerning three 
circles in a plane : 

The external centers of similitude of three circles of a plane are 
collinear. Any two internal centers of similitude are always collinear 
with one of the external centers. 

Fia. 2. 

To prove this, let us consider three spheres whose projections are 
the three given circles in the plane. Any two of these spheres admit 
of an internal and external common tangent cone. Thus, designating 
the centers of the spheres by Ci, C2, C3, we obtain three external and 
three internal tangent cones whose respective vertices E12, E23, E31, 
and I12, I23, Isi, are coplanar, the plane P passing through Ci, C2, C3. 
The three spheres and the six cones have the same common tangent 
planes and these are 2 by 2 symmetrical with respect to P. Any 
two symmetrical planes ( there are eight common tangent planes ) are 
common tangent planes to the three spheres and to three of the 
tangent cones. 

The three vertices of these cones are therefore necessarily collinear. 
We have therefore the result that the six vertices of the common 
tangent cones are coplanar and are 3 by 3 situated in straight lines; 
i. e., they form a complete quadrilateral. 

The collinear groups are, fig. 2 : 

E12 E12 E23 E31 

E23 I28 I31 1 12 

E31 I31 1 12 I23 



Fig. 3. 

Any orthographic or central projection of this configuration leads 
immediately to the original proposition. We may conversely use re- 
lations in a plane to establish propositions in space. Take, for 
instance, four circles in a plane and construct their external centers 
of similitude (fig. 3). It is found that they form a complete quadri- 
lateral, whose six points are 3 by 3 collinear. The four circles may 
be considered as projections of spheres in space. Here the collin- 
earity of the vertices of corresponding common tangent cones still 
exists, and we have therefore the theorem : 

The six external centers of similitude of any four spheres in space 
are coplanar. 

Similar propositions hold for the internal centers. 


In most text-books on descriptive geometry no attention is paid 
to certain geometrical principles which, as it will appear, form the 
base for nearly all projective constructions. It is also remarkable 
how easy these principles and relative propositions may be derived 
from exact intuition in space. From this point of view the necessity 
of including certain geometrical propositions in a course on descrip- 
tive geometry is imperative. I shall illustrate the points in question 
by the treatment of affinity of figures* (homology with an infinite 

* Fiedler: Geometrie der Lage, vol. 1, pp. 1-115. 



/ Fig. 4. 

The horizontal and vertical projections of any point in the bisecting 
plane of the second and fourth angle coincide. From this it follows 
that the projections of the piercing point of any line with the bisect- 
ing plane, which shall be designated by U, coincide, and are conse- 
quently obtained as the point of intersection of the two projections of 
the straight line. Let 1' and 1" be the projections of a line 1 and Di 
their point of intersection. In a similar manner, we designate by Ai, 

Bi, Ci, ; ai, bi, ci, the coinciding projections of points A, B, 0, 

. . . .and lines a, b, c, . . . . in U. As two points determine a straight 
line and as the projections of a point in U coincide, it follows that the 
projections of any line in U coincide also. Any plane P with the 
traces ti and t2 intersects U in a line u, whose projections coincide in 
ui. Evidently the traces ti, t2 and the lines u and ui meet in the same 
point T of the ground line. The line u is therefore determined if 
another point is known. To find such a point, assume any straight 
line 1 in the plane P, figure 4, and construct its intersection L witli 
U. The straight line connecting T with L, or in the projections T 
with Li, is the required line. Every line in the plane P intersects the 
line u ; hence its projections meet in a point of the line ui. From 
this it is seen that the projections of points, lines and figures in a 
plane P are related by the two laws : 

1. Corresponding projections of lines meet in points of a fixed 
straight line (axis of affinity). 

2. Corresponding projections of points are situated in parallel 


These are precisely the laws dominating the affinity of figures in a 
plane, which itself is a special case of homology (when the center is 
infinitely distant ) . The propositions which we have established in 
connection with homologous triangles may be specialized for triangles 
related by affinity and their proof does not present the slightest diffi- 
culty. In fact, affinity results from homology by considering a 
triangular prism instead of a triangular pyramid, and it is clear that 
our previous reasoning applies also to this case. 

I shall now mention two metrical specializations of affinity. Assume 
two triangles ABC and A'B'C related by affinity and assume that the 
parallel lines AA', BB', CC be perpendicular to the axis of affinity, 
then the ratios between the distances of corresponding points from 
the axis of affinity are equal and the proposition holds : 

The areas of the two triangles are to each other as the distances of 
corresponding points from the axis of affinity. 

If this ratio is unity then their areas are equal and the triangles 
are in axial symmetry. 

If AA', BB', CC are parallel to the axis of affinity then the areas 
of ABC and A'B'C are equal. 

This is a case which is never considered in plane geometry. 

I might extend this subject still further, but I hope the pre- 
vious treatment will be sufficient to show in what a simple and effect- 
ive manner, and without losing much time, important geometrical 
propositions may be obtained from the study of elementary de- 
scriptive geometry. It is hardly necessary to point out that these 
propositions are conversely the most efficient and rapid means to make 
projections of plane figures. 

Suppose, for instance, that the horizontal projection of a plane fig- 
ure A, B, C, ..., the vertical projection C" of C and the line of inter- 
section u of the plane of ABC... with U be given. The vertical 
projection A"B"C"... may be constructed by the previous principles 
alone. Thus, to find B", connect B'C and prolong to the intersec- 
tion with m; join the latter point to C", and from B' draw a perpen- 
dicular to the ground line. Where this perpendicular intersects the 
last line, is the required vertical projection B" of B. By this method 
three lines are sufficient to find a required point, while the ordinary 
method by means of the traces of the plane requires four or five 
( two parallels and two perpendiculars in one case, and two connecting 
lines and three perpendiculars in the other). The same principle 
may be applied to find the true shape of a plane figure from its projec- 
tions ; three lines give a point of the rabatted figure. This method is, 
therefore, also in the line of Lemoin's Gr6om6trographie,* where fig- 

*M. B. Lemoia : Principas de la Geometrograpuie, Archiv der Mathematik and Physik, 
vol. 1, 1901, p. 99. 




ures are constructed and investigated with reference to the greatest 
attainable simplicity. I can say from experience that there is hardly 
a subject in descriptive and advanced plane geometry in which the 
student takes more real interest than in this method of presenting the 
construction of plane figures in descriptive geometry, and of intro- 
ducing propositions of projective geometry. It seems to me the most 
natural way to higher geometry. 


As in the previous chapter, I shall start from elementary construc- 
tions in descriptive geometry and through the study of perspective 
gradually arrive at the most general expression of collineation. 

A central projection, or a perspective, is determined by the plane of 
projection (pictorial plane) and the center (eye) . Assuming the plane 
of the paper as the plane of projection and any point in space as the 
center, it is possible to construct the perspective of any figure in space 
in this plane. The center can be most easily located by a circle in 
the plane of projection. The radius of this circle is the distance of 
the center from the plane and the center of the circle is the ortho- 
graphic projection of the center of projection upon the plane of projec- 
tion. This circle has been introduced into geometry by Professor 
Fiedler, of Zurich, and he calls it distance circle (distanzkreis).* 

Fig. 5. 

Let II' be the plane of projection and II an arbitrary plane, whose 
projection upon II' shall be made from a center C. Let S be the line 
of intersection of II' and II, fig. 5. To obtain the projection of any 
point P' of any point P in II, connect P with the center ; then the 
point of intersection of this connecting line (indefinitely produced) 

*Darstellende Geometrie, vol. 1, 1883. It mnst be mentioned that Courinery already uses a 
"cercle a distance" in his Geometrie Perspective, Paris, 1828. 


with II' is the required point P'. In a similar manner the projection 
1' of a line 1 in II is obtained as the line of intersection of the plane 
passing through C and 1 with II'. From this construction the follow- 
ing fundamental laws are immediately clear, and are in fact a mere 
restatement of the laws of homologous triangles previously considered. 

To every point of II corresponds one and only one point of II', and 
conversely, and both points are on the same ray through C. 

To every straight line of II corresponds a straight line of II', and 
conversely ; and both lines meet at the same point of s ( holds for any 

To the infinite line q of II corresponds a line q' of II', which is 
parallel to s. Conversely, to the infinite line r' of II' corresponds a line 
r parallel to s. 

The plane II is usually determined by its trace s in II' and either 
of the lines r and q'. If a straight line 1 in II is given, intersecting 
the trace s in S', the corresponding line 1' is obtained by drawing a 
line through C parallel to 1 and marking its point of intersection Q' 
with q'. It is evident that Q' is the projection of the infinitive point 
of 1, and the projection of 1 consequently passes through S' and Q'. 
Another way is to produce 1 till it intersects r in K and to join C with 
R. The line through S parallel to OR is the required projection 1' 
of 1. From the figure, it is seen that ORSQ' is a parallelogram and 
thatPS:PR = FS:CR. 

Fig. 6. 

The planes through parallel to II and II' form a space of a paral- 
lelepipedon. Keeping II' fixed it is possible by rotations about s and 


q' as axes to rabatte the planes through 0, and II into II' without 
changing the figures situated in these planes. After the motion there 
is still CR=and || Q'S and SP' = SP', so that the distances PR and 
PS remain also unchanged. From this it follows that after the mo- 
tion P' and the rabatted position of P lie on a ray through the re- 
volved position of C. The laws expressing the relations between the 
revolved and projected figure are therefore the same as those between 
the figure in space (II) and its projection II'. After the rabatte- 
ment, fig. 5 assumes the form of fig. 6. Here 1 and 1' are the two cor- 
responding lines which with s and SC form a pencil of four rays 
through S. As CP and CQ intersect this pencil there is 

The value of (CMQ'Q) is ^-, = ^ = k, ray ; i. e., entirely independ- 
ent of the position of 1, 1', and OP. Thus, drawing any ray through C 
and intersecting s in S', any two points P and P' on this ray of the 
central projection form a constant ratio with C and S'. For all pos- 
sible pairs P and P' of corresponding points 

(CSPP') = constant. 
The different cases of central projection may be classified according 
to the position of the center of projection and the value of the con- 
stant k.* 

Laying any Carterian system of coordinates through C, and desig- 
nating the coordinates of any pair of corresponding points P and P' 
by x, y, and x', y', respectively, it is an easy matter to derive from the 
figure the general form of the relation existing between these coordi- 
nates : 

„'_ as 1 

x — dx + ey + f I 

^ (1) 

J dx + ey + f J 

where — is the constant k of the projection. 

Formulas (1) represent the transformation of a point P into P', 
called perspective. 

The equation of the line r is dx + ey + f = o, and its correspond- 
ing line r' is infinitely distant. To the line q(x = oo, y — co) corre- 
sponds the line q' with the equation dx'+ ey' — a = o. The axis s 
is obtained by putting x'= x, y'= y, and its equation is dx + ey + f 
— a — o. Comparing the equations of the lines r, s, q', and their dis- 
tances from O, it is found that 

f— a f a 

Vtf+e* v/d'+e 2 i/d s +e 2 ' 

*Pi«dler, loo. oit., p. 95. 


as it also appears from the figure. The deduction of formulas (1) 
from an actual perspective construction has the great advantage that 
all results gained from the analytical discussion may easily be inter- 
preted constructively and geometrically. It is noticed that a per- 
spective transformation depends upon three essential constants, i. e., 
if its center is fixed. This is equivalent to saying that the axis of 
perspective can be chosen in a doubly infinite number of ways, and 
the constant k in a singly infinite number. If to the perspective 
transformation P, as given by (1), we apply a dilation D, defined by 

x"=dx', y"=y', (2) 

and which may be considered as a special case of perspective where 
the center is infinite in a direction perpendicular to s. From this it 
is seen that the combination of a perspective and dilation ( PD ) may 
be expressed by 

„ _ ax' 1 

dx' + ey' + f 

V (3) 


J "dx'+ey' + f 

Applying to this transformation consecutively a transformation by 
equal areas (A), defined by 

x'"=x" + ky" 


and which may also be considered as a special case of perspective, in 
which C is infinitely distant in a plane perpendicular to the bisecting 
plane of II and II'; then a transformation (T), defined by 

X (4) = x '" + p 

y (4 > = y"' + q ' ' (5) 

and finally a rotation ( R ), 

x (5) =nx (4)_ my (4) 

y(5) =nx (4)+ my (4) I { -' J ' 

(m 2 + n 2 = l), 
we arrive at a transformation of the form 

, _ ax + by + c ; 1 

dx + ey + f 
> _ gx + hy + j 


' — dx + ey + f 

This transformation is characterized by eight independent con- 
stants, and is called, as is well known, a projective transformation, or 
collineation. It may be considered as the result of the combined 


(PDATR), (8) 

which are determined by 3, 1, 1, 2, 1 parameters, respectively. It is 


also well known that the latter are subgroups of the general project- 
ive group. Prom the constructive study of collineation we have thus 
arrived at the conception of the continuous projective groups of trans- 
formation. The method which we have followed makes it again pos- 
sible to follow the train of reasoning in the discussion of groups by 
illustrative constructions. It may, of course, be extended to space. 


A well-arranged parallelism of descriptive, synthetic and analytic 
methods in organic connection, a method chiefly cultivated by Fied- 
ler, seems to be most valuable for a rapid introduction into the fields 
of higher geometry. The introduction of critical discussion concern- 
ing the foundations of geometry into elementary treatises has a 
tendency to confuse the student. The establishment of the funda- 
mental principles of projective geometry independent of metrical 
relations or of the eleventh axiom of Euclid may follow an introduc- 
tion as outlined in this paper, v. Standt's construction, Fiedler's 
projective coordinates, Caley's and Klein's absolute geometry, or non- 
Euclidian geometry, must form indispensable parts of such an ad- 
vanced study. In the method followed by us, and which is partly, 
also, that of Poncelet, Steiner, and Charles, the projective properties 
of the circle are easily established and transferred to conies by per- 
spective. It is, however, necessary to show that all curves of the 
second order defined as products of projective ranges and pencils, or 
analytically by equations of the second degree, are conies. There is 
no difficulty in doing this. 

Descriptive analytic methods are also of invaluable service for the 
study of congruences and complexes of rays and for higher geometry 
in general. In this respect I may mention the treatment of linear 
complexes, the congruence of bisecants of a twisted cubic, of the 
"Null system," by descriptive methods, and their elegant representa- 
tion by certain partial' differential equations.* 

There is one branch of mathematics which is rarely mentioned in 
connection with projective geometry, namely, kinematics. In the 
hands of Penucellier, Kempe, Sylvester, Hart, and, in recent times, 
especially by Professor Koenigs, of Paris, kinematics has rendered 
valuable services to modern geometry. Starting from the beautiful 
theorem f that every plane and twisted algebraic curve and every alge- 
braic surface may be described by a linkage, Koenigs invented a plani- 
graph, and quite recently, also, a link-motion perspectivograph, real- 
izing collineation. A short treatment of these interesting linkages 
would form a valuable addition to any text- book on projective geometry. 

•See my paper "On the Congruences of Rajs (3,1) and (1,3)," Annals of Mathematics; 
also, S. Lie: Geometrie der Beruhrungs-transformationen, vol. I, p. 326. 

t Koenigs: Lecons de Cinematique, Paris, 1897, pp. 271, 297, 305.